Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13534 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,15,19,22,37,11,18,14,21,12,28,13,20,27,31)(9,17,24,34,30,32,23,33,26,36,10,16,25,35,29,38);(3,4)(5,6)(7,9,22,25,14,26,13,30)(8,16,20,38,21,34,37,33)(10,15,23,11,24,12,29,27)(17,31,32,28,36,18,35,19);(4,6)(7,10)(8,17)(9,15)(11,25)(12,26)(13,29)(14,24)(16,31)(18,34)(19,33)(20,32)(21,36)(22,23)(27,30)(28,38)(35,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406735);